In this article we investigate the job Sequencing and tool Switching Problem (SSP), a NPNP-hard combinatorial optimization problem arising from computer and manufacturing systems. Starting from the results described in Tang and Denardo (1987), Crama et al. (1994) and Laporte et al. (2004), we develop new integer linear programming formulations for the problem that are provably better than the alternative ones currently described in the literature. Computational experiments show that the lower bounds obtained by the linear relaxation of the considered formulations improve, on average, upon those currently described in the literature and suggest, at the same time, new directions for the development of future exact solution approaches.